Question

A customized option on a stock pays the option holder (S - K)^2 where S is the

stock price at maturity, one week from now, and K is $54. The current stock price is $50.

There are only two possibilities one week from now, that the stock is $59 with probability

70%, or that it's $47 with probability 30%. Taking dividends and interest rates to both

be zero, compute the price of this derivative using a one-step binomial tree. Compute the

derivative price two ways:

1) by setting up a delta-hedged portfolio and working out the cost

2) by determining risk-neutral probabilities and applying them.

Answer 1

Data hedging by using John Hull method:

Suppose we want to value an option giving the right to buy a stock for the strike price K is $54 (the price in the contract) in one week (expiry date T). The current stock price S is $50.

We make a very simplifying assumption which in reality is not satisfied:

In one week the stock price will be either $59 or $47. With the probability of 70% and 30%.

However, at expiry date, we know the value of the option:

If the stock price goes up (to $59), the payoff − K is $1;

If the stock price goes down (to $49), then the option is worthless, so the value is 0.

A portfolio with one stock and ∆ shares

Look at a portfolio which is generated at time 0 by buying ∆ many shares of a stock (long position) and selling one call option of the stock (short position).

If f is the price of the option the portfolio at time 0 has value

                                            50∆ − f

After one week the value of this portfolio can be computed:

If the stock prices moves from $50 to $59 then the value of the shares is $59∆ and the value of the option is $1, so the total value of the portfolio is

                                            59∆ − 1

(in this case our contract partner has the right to buy from us one stock at the strike price $58, so we have to give him $1); If the stock prices moves from $50 to $47 then the value of the shares is $47∆ and the value of the option is zero, so the total value of the portfolio 47∆.

Strategy: delta hedging:

The idea is now to choose ∆ such that the value of the portfolio in both cases (stock price up or down) is the same:

So we require:

                              59∆ − 1 = 47∆, so ∆ = 0.083

Control: in one week

If = $59, then the portfolio is worth 59 * 0.083 − 1 = $ 3.89 or 3.90 Portfolio

If = $47, then the portfolio is worth 47 * 0.083 = $ 3.90 Portfolio

Definition:

The delta of an option is the number ∆ of shares we should hold for one option (short position) in order to create a riskless hedge: so after maturity time T the value of the portfolio containing ∆ share and selling one call option is for both cases ST > K and ST ≤ K the same. The construction of a riskless hedge or Neutral Hedging is called delta hedging of portfolio.